Yes you can.
Have proceedings for anything been issued yet? The divorce itself? For financial orders? If nothing at all has been issued, you will have a bigger task to do, because you'd have to issue the divorce petition as well.
Assuming Form A has already been issued at court, start by looking at your Form A. Check that the box has been ticked asking for maintenance pending suit. If it hasn't, print off a new Form A and tick that box as well as the others (for property adjustment, etc etc). Your application will be for permission to amend the Form A as well as to claim MPS.
Next, you need a witness statement. Start by drawing up a schedule of current monthly income and outgoings for yourself. Get a few examples of suitable properties to show what you'd need in terms of rental. Copy the letters from the letting agents which show that your rent will need to be paid up in advance (or get letter to show this).
In your witness statement you need to state what you are asking for (ie a deposit and 12 months' rent in advance, or whatever it may be). You can refer to it as capitalised monthly periodical payments, but make it clear that you need it now because you have nowhere to move to and are unable to secure a rental property on your own. You also need to state where the money will come from - so explain why you think ex has ££ savings in his name and can pay this immediately.
It is important to remember that MPS cannot be used to get an advance on the capital settlement. For example, you can't say ,"I believe I will be awarded £50k from the sale of the house so I'd like part of it in advance". MPS is only for money that you need to maintain yourself (and the DC) until the final order is made. That's why I'm suggesting you describe it as capitalised maintenance, because it refers to monthly payments, but in your case you do actually have to have it up front.
Send it all off to ex-p, and as I say, give him 48 hours to respond. If he doesn't come through, make 3 copies of everything & send them to the court with a covering letter making it clear that you need a very urgent hearing as you are likely to lose your home within 14 days. It might be as well to ring the court first to check what you have to do regarding court fees, and you can check about getting an urgent hearing.
Do remember to tell your solicitor if you are doing this, and decide whether you are going to keep them on or not.